Chern-Simons Gauge Theory Research Articles

Some lower dimensional quantum field theories reduced from Chern-Simons gauge theories

We study symmetry reductions in the context of Euclidean Chern-Simons gauge theories to obtain lower dimensional field theories. Symmetry reduction in certain gauge theories is a common tool for obtaining explicit soliton solutions. Although pure Chern-Simons theories do not admit solitonic solutions, symmetry reduction still leads to interesting results. We establish relations at the semiclassical regime between pure Chern-Simons theories on S3 and the reduced quantum field theories, based on actions obtained by the symmetry reduction of the Chern-Simons action, spherical symmetry being the prominent one. We also discuss symmetry reductions of Chern-Simons theories on the disk, yielding a topological theory in two dimensions, which signals a curious relationship between symmetry reductions and the boundary conformal field theories. Finally, we study the Chern-Simons-Higgs instantons and show that under certain circumstances, the reduced action can formally be viewed as the action of a supersymmetric quantum mechanical model. We discuss the extent to which the reduced actions have a fermionic nature at the level of the partition function. Published by the American Physical Society 2024

Self-gravitating solutions in Yang–Mills–Chern–Simons theory coupled to 3D massive gravity

We study self-gravitating solutions of three-dimensional massive gravity coupled to the Yang–Mills–Chern–Simons gauge theory. Among these, there is a family of asymptotically Warped-Anti de Sitter black holes that come to generalize previous solutions found in the literature and studied in the context of WAdS3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document}/CFT2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}. We also present self-gravitating solutions to three-dimensional Einstein–Yang–Mills theory, as well other self-gravitating solutions in the presence of higher-curvature terms.

Homological aspects of topological gauge-gravity equivalence

In the works of Achúcarro and Townsend and also by Witten, a duality between three-dimensional Chern–Simons gauge theories and gravity was established. In all cases, the results made use of the field equations. In a previous work, we were capable of generalizing Witten’s work to the off-shell cases, as well as to the four-dimensional Yang–Mills theory with de Sitter gauge symmetry. The price we paid is that curvature and torsion must obey some constraints under the action of the interior derivative. These constraints implied on the partial breaking of diffeomorphism invariance. In this work, we first formalize our early results in terms of fiber bundle theory by establishing the formal aspects of the map between a principal bundle (gauge theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism invariance. Then, we study the effect of the constraints on the homology defined by the interior derivative. The main result is the emergence of a non-trivial homology in Riemann–Cartan manifolds.

Superspin chains solutions from 4D Chern-Simons theory

As a generalisation of the correspondence linking 2D integrable systems with 4D Chern-Simons (CS) gauge theory, superspin chains are realized by means of crossing electric and magnetic super line defects in the 4D CS with super gauge symmetry. The oscillator realization of Lax operators solving the RLL relations of integrability is obtained in the gauge theory by extending the notion of Levi decomposition to Lie superalgebras. Based on particular 3-gradings of Lie superalgebras, we obtain graded oscillator Lax matrices for superspin chains with internal symmetries given by A(m − 1 | n − 1), B(m | n), C(n) and D(m | n).

W(0,b) algebra and the dual theory of 3D asymptotically flat higher spin gravity

Bondi-Metzner-Sachs (BMS) algebra in three spacetime dimensions can be deformed into a two parameter family of algebra known as W(a,b) algebra. For a=0, we show that other than W(0,−1), no other W(0,b) algebra admits a nondegenerate bilinear and thus one cannot have a Chern-Simons gauge theory formulation with them. However, they may appear in a three-dimensional gravity description, where we also need to have a spin 2 generator, that comes from the (a=0,b=−1) sector. In the present work, we have demonstrated that the asymptotic symmetry algebra of a spin-3 gravity theory on flat spacetime has both the W(0,−1) and W(0,−2) algebras as subalgebras. We have also constructed a dual boundary field theory for this higher spin gravity theory by using the Chern-Simons/Wess-Zumino-Witten correspondence. Published by the American Physical Society 2024

Extended phase space in general gauge theories

In a recent paper, it was shown that in diffeomorphism-invariant theories, the symplectic vector fields induced by spacetime diffeomorphisms are integrable if one introduces an extended phase space. In this paper we extend the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs (φ,Φ), where φ is a Lie algebroid morphism and Φ is a field configuration in the non-extended sense. Starting from this data, we outline a very powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons gauge theories and display its usefulness by re-computing the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories.

Noncommutative $$SO(2,3)_{\star }$$ gauge theory of gravity

Topological gravity (in the sense that it is metric-independent) in a 2n-dimensional spacetime can be formulated as a gauge field theory for the AdS gauge group $$SO(2,2n-1)$$ by adding a multiplet of scalar fields. These scalars can break the gauge invariance of the topological gravity action, thus making a connection with Einstein’s gravity. This review is about a noncommutative (NC) star-product deformation of the four-dimensional AdS gauge theory of gravity, including Dirac spinors and the Yang–Mills field. In general, NC actions can be expanded in powers of the canonical noncommutativity parameter $$\theta$$ using the Seiberg–Witten map. The leading-order term of the expansion is the classical action, while the higher-order $$\theta$$ -dependent terms are interpreted as new types of coupling between classical fields due to spacetime noncommutativity. We study how these perturbative NC corrections affect the field equations of motion and derive some phenomenological consequences, such as NC-deformed Landau levels of an electron. Finally, we discuss how topological gravity in four dimensions (both classical and noncommutative) appears as a low-energy sector of five-dimensional Chern–Simons gauge theory in the sense of Kaluza–Klein reduction.

Spin-flux attachment by dimensional reduction of vortices

The description of a system of vortices in terms of dual fields provides a window to new phases of the system. It was found recently that dualizing a 3+1-d boson-fermion system leads to a system of fermions and vortices interacting via a 2-form field through a non-local term. Here we explore some consequences of such an interaction when the degrees of freedom of the system are confined to a 2+1-d space-time. In particular, we show that the vortices in the 2+1-d system are attached to the fermions via their non-zero spin magnetic moment in a way similar to the phenomenon of flux attachment in Chern-Simons gauge theory coupled to matter. We also show that such flux attached particles exhibit fractional statistical behaviour like anyons. Thus our model provides a realization of anyons without Chern-Simons theory.

How planar superconductors cure their infrared divergences

Planar superconductors, emerging in thin films with thickness comparable to the superconducting coherence length, differ crucially from their bulk counterparts. Coulomb interactions between charges are logarithmic up to distances comparable to typical sample sizes and the Anderson-Higgs mechanism is ineffective to screen the infrared divergences of the resulting (2+1)-dimensional QED because the Pearl length screening the vortex interactions is also typically larger than the sample size. As a result, the system decomposes into superconducting droplets with the typical size of order of superconducting coherence length. We show that two possible phases of the film match the two known mechanisms for curing the (2+1)-dimensional QED infrared divergences, either by generating a mixed topological Chern-Simons mass or by magnetic monopole instantons. The former mechanism works in superconductors, the latter one governs mirror-dual superinsulators. Planar superconductors are thus described by a topological Chern-Simons gauge (TCSG) theory that replaces the Ginzburg-Landau model in two dimensions. In the TCSG model, the Higgs field is absent. Accordingly, in planar superconductors Abrikosov vortices do not form, and only Josephson vortices without normal core do exist.

The Hilbert space of large N Chern-Simons matter theories

We demonstrate that the known expressions for the thermal partition function of large N Chern-Simons matter theories admit a simple Hilbert space interpretation as the partition function of an associated ungauged large N matter theory with one additional condition: the Fock space of this associated theory is projected down to the subspace of its quantum singlets i.e. singlets under the Gauss law for Chern-Simons gauge theory. Via the Chern-Simons / WZW correspondence, the space of quantum singlets are equivalent to the space of WZW conformal blocks. One step in our demonstration involves recasting the Verlinde formula for the dimension of the space of conformal blocks in SU(N)k and U(N)k,k′ WZW theories into a simple and physically transparent form, which we also rederive by evaluating the partition function and superconformal index of pure Chern-Simons theory in the presence of Wilson lines. A particular consequence of the projection of the Fock space of Chern-Simons matter theories to quantum (or WZW) singlets is the ‘Bosonic Exclusion Principle’: the number of bosons occupying any single particle state is bounded above by the Chern-Simons level. The quantum singlet condition (unlike its Yang-Mills Gauss Law counterpart) has a nontrivial impact on thermodynamics even in the infinite volume limit. In this limit the projected Fock space partition function reduces to a product of partition functions, one for each single particle state. These single particle state partition functions are q-deformations of their free boson and free fermion counterparts and interpolate between these two special cases. We also propose a formula for the large N partition function that is valid for arbitrary finite volume of the spatial S2 and not only at large volume.

Fractonic order in infinite-component Chern-Simons gauge theories

2+1D multi-component $U(1)$ gauge theories with a Chern-Simons (CS) term provide a simple and complete characterization of 2+1D Abelian topological orders. In this paper, we extend the theory by taking the number of component gauge fields to infinity and find that they can describe interesting types of 3+1D "fractonic" order. "Fractonic" describes the peculiar phenomena that point excitations in certain strongly interacting systems either cannot move at all or are only allowed to move in a lower dimensional sub-manifold. In the simplest cases of infinite-component CS gauge theory, different components do not couple to each other and the theory describes a decoupled stack of 2+1D fractional Quantum Hall systems with quasi-particles moving only in 2D planes -- hence a fractonic system. We find that when the component gauge fields do couple through the CS term, more varieties of fractonic orders are possible. For example, they may describe foliated fractonic systems for which increasing the system size requires insertion of nontrivial 2+1D topological states. Moreover, we find examples which lie beyond the foliation framework, characterized by 2D excitations of infinite order and braiding statistics that are not strictly local.

Notes on Chern–Simons perturbation theory

We give a detailed introduction to the classical Chern–Simons gauge theory, including the mathematical preliminaries. We then explain the perturbative quantization of gauge theories via the Batalin–Vilkovisky (BV) formalism. We then define the perturbative Chern–Simons partition function at any (possibly non-acylic) reference flat connection using the BV formalism, using a Riemannian metric for gauge fixing. We show that it exhibits an anomaly known as the “framing anomaly” when the Riemannian metric is changed, that is, it fails to be gauge invariant. We explain how one can deal with this anomaly to obtain a topological invariant of framed manifolds.

Structure of Chern-Simons scattering amplitudes from topological equivalence theorem and double-copy

We study the mechanism of topological mass-generation for 3d Chern-Simons (CS) gauge theories, where the CS term can retain the gauge symmetry and make gauge boson topologically massive. Without CS term the 3d massless gauge boson has a single physical transverse polarization state, while adding the CS term converts it into a massive physical polarization state and conserves the total physical degrees of freedom. We formulate the mechanism of topological mass-generation at S-matrix level. For this, we propose and prove a Topological Equivalence Theorem (TET) which connects the N-point scattering amplitude of the gauge boson’s physical polarization states ( {A}_{mathrm{P}}^a ) to that of the transverse polarization states ( {A}_{mathrm{T}}^a ) under high energy expansion. We present a general 3d power counting method on the leading energy dependence of the scattering amplitudes in both topologically massive Yang-Mills (TMYM) and topologically massive gravity (TMG) theories. With these, we uncover a general energy cancellation mechanism for N -gauge boson scattering amplitudes which predicts the cancellation E4→ E4−N at tree level. Then, we compute the 4-gauge boson amplitudes of {A}_{mathrm{P}}^a -states and {A}_{mathrm{T}}^a -states, with which we explicitly demonstrate the TET and establish such energy cancellations for N = 4. We further extend the double-copy approach to reconstruct the massive 4-graviton amplitude of TMG from the massive 4-gauge boson amplitude of TMYM. With these, we uncover striking large energy cancellations in the 4-graviton amplitude: E12→ E1, and establish its correspondence to the leading energy cancellation E4→ E0in the 4-gauge boson amplitude of TMYM.

Fractionally charged BPS particles in M-string on 3D orbifolds

In this paper, we study the M-string realization of chiral [Formula: see text]-super-conformal field theory in 6 dimensions and its orbifold compactification down to three-dimensional (3D). We analyze its fractionally charged BPS particle spectrum in connection with effective 3D Chern–Simons gauge theory and the supersymmetric fractional quantum Hall effect in [Formula: see text] dimensions. We construct the set of underlying fractionally charged BPS particles in the ground state of the compactified M string and find that it contains 144 BPS states that are generated by four basic quasi-particles (two bosonic-like and two fermionic like) and their CPT conjugate. Two representations of the gauge bosons and the gauginos as condensates of the basic quasiparticles are found and explicit realizations are also given. Other features concerning generalizations are also discussed.

A cohomological obstruction in higher-dimensional Chern–Simons gauge theories

We study a set of cohomology classes which emerge in the cohomological formulations of the calculus of variations as obstructions to the existence of (global) solutions of the Euler–Lagrange equations of Chern–Simons gauge theories in higher dimensions [Formula: see text].

Poisson–Lie Groups and Gauge Theory

We review Poisson–Lie groups and their applications in gauge theory and integrable systems from a mathematical physics perspective. We also comment on recent results and developments and their applications. In particular, we discuss the role of quasitriangular Poisson–Lie groups and dynamical r-matrices in the description of moduli spaces of flat connections and the Chern–Simons gauge theory.

The Hamilton–Jacobi analysis for higher-order Maxwell–Chern–Simons gauge theory

By using the Hamilton–Jacobi [HJ] framework, the higher-order Maxwell–Chern–Simons theory is analyzed. The complete set of HJ Hamiltonians and a generalized HJ differential are reported, from which all symmetries of the theory are identified. In addition, we complete our study by performing the higher-order Gitman–Lyakhovich–Tyutin [GLT] framework and compare the results of both formalisms.

Faddeev–Jackiw quantization of the higher derivative Chern–Simons gauge theory in the first-order formalism

We analyse the physical constraints of the higher derivative Chern–Simons gauge model by means of Faddeev–Jackiw symplectic approach in the first-order formalism. Within such framework, we systematically determine the zero-mode structure of the corresponding symplectic matrix. In addition, we calculate the Faddeev–Jackiw quantum brackets by choosing appropriate gauge-fixing conditions and evaluate the determinant of the non-singular symplectic matrix as well as the transition-amplitude. Finally, we present a detailed Hamiltonian analysis using Dirac–Bergmann algorithm method and show that the Dirac brackets coincide with the FJ brackets when all the second-class constraints are treated as zero equations.

Vassiliev invariants for flows via Chern–Simons perturbation theory

The perturbative expansion of Chern–Simons gauge theory leads to invariants of knots and links, the so-called finite type invariants or Vassiliev invariants. It has been proved that at any order in perturbation theory the superposition of certain amplitudes is an invariant of that order. Bott–Taubes integrals on configuration spaces are introduced in the present context to write Feynman diagrams at a given order in perturbation theory in a geometrical and topological framework. One of the consequences of this formalism is that the resulting amplitudes are rewritten in cohomological terms in configuration spaces. This cohomological structure can be used to translate Bott–Taubes integrals into Chern–Simons perturbative amplitudes and vice versa. In this paper, this program is performed up to third order in the coupling constant. This expands some work previously worked out by Thurston. Finally we take advantage of these results to incorporate in the formalism a smooth and divergenceless vector field on the [Formula: see text]-manifold. The Bott–Taubes integrals obtained are used for constructing higher-order average asymptotic Vassiliev invariants extending the work of Komendarczyk and Volić.

On quantum Hall effect, Kosterlitz-Thouless phase transition, Dirac magnetic monopole, and Bohr–Sommerfeld quantization

We addressed quantization phenomena in open systems and confined motion in low-dimensional systems, as well as quantized sources in 3-dimensions. The thesis of the paper is that if we simply cast the Bohr–Sommerfeld (B-S) quantization condition as a U(1) gauge theory, like the gauge field of Chern-Simons gauge theory or as in topological band theory (TBT) of condensed matter physics in terms of Berry connection and Berry curvature to make it self-consistent, then the quantization method in all the physical phenomena treated in this paper are unified in the sense of being traceable to the self-consistent B-S quantization. These are the stationary quantization of due to oscillatory dynamics in compactified space and time for steady-state systems (e.g., particle in a box or torus, Brillouin zone, and Matsubara time zone or Matsubara quantized frequencies), and the quantization of sources through the gauge field. Thus, the self-consistent B-S quantization condition permeates the quantization of integer quantum Hall effect (IQHE), fractional quantum Hall effect (FQHE), the Berezenskii-Kosterlitz-Thouless vortex quantization, Aharonov–Bohm effect, the Dirac magnetic monopole, the Haldane phase, contact resistance in closed mesoscopic circuits of quantum physics, and in the monodromy (holonomy) of completely integrable Hamiltonian systems of quantum geometry. In transport of open systems, we introduced a novel phase-space derivation of the quantized conductance of the IQHE based on nonequilibrium quantum transport and lattice Weyl transform approach.